Too little info?

Since ${ x }^{ 4 }-{ y }^{ 4 }=2009$ , it follows that $({ x }^{ 2 }+{ y }^{ 2 })({ x }^{ 2 }-{ y }^{ 2 })=2009$ .

But ${ x }^{ 2 }+{ y }^{ 2 }=49$ hence ${ x }^{ 2 }-{ y }^{ 2 }=\frac { 2009 }{ 49 } =41$ .

Subtracting gives $2{ y }^{ 2 }=8$ hence ${ y }^{ 2 }=4$ .

Since $y>0$ , $y=\boxed { 2 }$ .

$x^4 - y^4 = 2009$

$x^2 + y^2 = 49$

This can be solved with simultaneous equations by turning the second equation into one for $x^2$

$x^2 = 49 - y^2$

Since $x^4 = x^2 \cdot x^2$ we can substitute it into the top equation

$(49 - y^2)^2 - y^4 = 2009$

Expanding the bracket gives

$2401 - 98y^2 + y^4 - y^4 = 2009$

The $y^4$ s cancel out so we get the equation

$2401 - 98y^2 = 2009$

Rearranging this gives us

$98y^2 = 392$

Almost there, now we divide by $98$ to get

$y^2 = 4$

So that means that

$y = 2$

we have x^4 + y^4=2009 and x^2 + y^2=49 by expressing x in y in the second equation we have x^2=49-y^2...so let replace it in the first equation... with x^4=(x^2)^2, (49-y^2)^2 - y^4=2009 after developping and eliminating elements we finally find 98y^2=392 then y^2=4 because y>0 so y=2

x ^ 2 + y ^ 2 = 49 x ^ 2 = 49 - y ^ 2 replacement (49 - y ^ 2) ^ 2 - y ^ 4 = 2009 (2401 - 98y ^2 + y ^ 4) - y ^ 4 = 2009 y ^ 2 = (2009 - 2401) / (-98) = 4 y = 2

x ^ 2 + 4= 49 x ^ 2 = 45 x ~= 6;708

x^4 - y^4 =( x^2 + y^2)(x^2 - y^2).

2009 = 49(x^2 - y^2).

x^2 - y^2 = 41.

y^2 = 4.

y = 2.